Відповіді на вправи: 702

702. Знайдіть суму:

1) 1 + 2 + 4 + … + 1024.

b₁ = 1; b₂ = 2; q = $\frac{b_2}{b_1}$ = $\frac{2}{1}$ = 2.

b_n = b₁q^n–1;

1 • 2^n–1 = 1024; 2ⁿ • 2^–1 = 1024;

2ⁿ • $\frac{1}{2}$ = 1024; 2ⁿ = 2048;

2ⁿ = 2¹¹, n = 11.

S₁₁ = $\frac{1(2^{11}-1)}{2-1}$ = 2047.

2) 5 – 15 + 45 – … + 405.

b₁ = 5; b₂ = –15; q = $\frac{b_2}{b_1}$ = –$\frac{15}{5}$ = –3.

b_n = b₁q^n–1; 5(–3)^n–1 = 405;

(–3)ⁿ (–$\frac{5}{3}$) = 405; (–3)ⁿ = 243;

(–3)ⁿ = –35, n = 5.

S₅ = $\frac{5\left(\right(-3{)}^5-1)}{-3-1}$ = $\frac{5-244}{-4}$ = 305.